Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{r^2 - 5r - 24}{r^2 + 3r}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - 5r - 24}{r^2 + 3r} = \dfrac{(r - 8)(r + 3)}{(r)(r + 3)} $ Notice that the term $(r + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 3)$ gives: $x = \dfrac{r - 8}{r}$ Since we divided by $(r + 3)$, $r \neq -3$. $x = \dfrac{r - 8}{r}; \space r \neq -3$